The arguement here follows from the proof of [1]}. Let \(\mathbb {L}\in \mathcal {Q}\)  and \(s_\mathbb {L}=s-j\) . We assume that \(\Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{d-1})}=1\) . Lemma REF , together with Hölder's inequality, yields that
\(&&\int _{\mathbb {R}^d}\int _{\mathbb {R}^d}|K^*_s(x,\,y)||b_\mathbb {L}(y)||h(x)|dydx\\&&\quad \lesssim \int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\frac{|A_\mathbb {L}(x)-A_\mathbb {L}(y)|}{|x-y|^{d+1}}\phi _{s}(x-y)|b_\mathbb {L}(y)||h(x)|dx\\&&\qquad +\sum _{n=1}^d2^{-sd}\int _{\mathbb {R}^d}\int _{\mathbb {R}^d}|b_\mathbb {L}(y)||\partial _n A(x)-\langle \partial _n A\rangle _{\mathbb {L}}||h(x)|\chi _{{\mathbb {L}^s}}(x)\,dy\,dx\\&&\quad \lesssim j\Vert b_\mathbb {L}\Vert _{L^1(\mathbb {R}^d)}\inf _{x\in \mathbb {L}^*}Mh(x)+\sum _{n=1}^d\Big (\frac{1}{|\mathbb {L}^s|}\int _{\mathbb {L}^s}|\partial _n A(x)-\langle \nabla A\rangle _{\mathbb {L}^s}|^{u^{\prime }}dx\Big )^{\frac{1}{u^{\prime }}}\\&&\qquad \times \inf _{x\in \mathbb {L}^*}M_uh(x)\Vert b_\mathbb {L}\Vert _{L^1(\mathbb {R}^d)}\\&&\quad \lesssim ju^{\prime }|\mathbb {L}|\Vert b\Vert _{\dot{\mathcal {X}_1}(\mathcal {Q})}\Vert h\Vert _{\mathcal {Y}_u(\mathcal {Q})},\) 
where \(\mathbb {L}^s=B(y,2^{s+10})\) , and here we have used the fact that
\(\Big (\frac{1}{|\mathbb {L}^s|}\int _{\mathbb {L}^s}|\partial _n A(x)-\langle \partial _n A\rangle _{\mathbb {L}^s}|^{u^{\prime }}dx\Big )^{\frac{1}{u^{\prime }}}\lesssim u^{\prime },\) 
see [2]}. This completes the proof of Lemma REF .
